aa r X i v : . [ m a t h . C V ] F e b SEGRE-DEGENERATE POINTS FORM A SEMIANALYTIC SET
JIŘÍ LEBL
Abstract.
We prove that the set of Segre-degenerate points of a real-analytic subvariety X in C n is a closed semianalytic set. If X is coherent, it is a subvariety. More precisely,the set of points where the germ of the Segre variety is of dimension k or greater is aclosed semianalytic set in general, and for a coherent X , it is a real-analytic subvariety of X . For a hypersurface X in C n , the set of Segre-degenerate points, X [ n ] , is a semianalyticset of dimension at most n − . If X is coherent, then X [ n ] is a complex subvariety of(complex) dimension n − . Example hypersurfaces are given showing that X [ n ] need notbe a subvariety and that it also needs not be complex; X [ n ] can, for instance, be a real line. Introduction
Segre varieties are a widely used tools for dealing with real-analytic submanifolds in com-plex manifolds. Recently, there have been many applications of Segre variety techniques tosingular real-analytic subvarieties, and while the techniques are powerful, they have to beapplied carefully. It is tempting to cite an argument or result for submanifolds proving thesame result for subvarieties, but there are two things that can go wrong. First, the Segrevariety can be degenerate (of wrong dimension), and second, the variety itself may be notcoherent, and the Segre variety cannot be defined by the same function(s) at all points. Onecannot define Segre varieties with respect to the complexification at one point and expectit to give a well-defined Segre variety at all nearby points (germs have complexifications,but their representatives may not). One wrong statement that may have already made itinto the “folklore” is that the set of Segre degenerate points of a real hypersurface in C n isa complex-analytic subvariety. The result may seem obvious, but it is only true for coher-ent hypersurfaces. The set of Segre-degenerate points of a hypersurface is not only not acomplex-analytic subvariety in general, it need not even be a real-analytic subvariety, it is asemianalytic set. We give an example where it is not a subvariety, and also one where it isof odd real dimension.The idea of using Segre varieties is old, although the techniques in using them in CR geome-try were brought into prominence first by Webster [12] and Diederich–Fornæss [5]. For a goodintroduction in their use for submanifolds, see the book by Baouendi–Ebenfelt–Rothschild [2].They started to be used for singular subvarieties recently, see for example Burns–Gong [4],Diederich–Mazzilli [6], the author [9], Adamus–Randriambololona–Shafikov [1], Fernández-Pérez [7], Pinchuk–Shafikov–Sukhov [11], and many others. However, the reader should becareful that sometimes in the literature on singular subvarieties a Segre variety is defined withrespect to a single defining function and it is not made clear that it is then not well-definedif the point moves. Date : February 13, 2021.2020
Mathematics Subject Classification.
Key words and phrases.
Segre degenerate, Segre variety, semianalytic.The author was in part supported by Simons Foundation collaboration grant 710294.
A good reference for real-analytic geometry is Guaraldo–Macrì–Tancredi [8], and a goodreference for complex analytic subvarieties is Whitney [13].A real-analytic subvariety of an open U ⊂ C n is a closed subset X ⊂ U defined locallyby the vanishing of real-analytic functions. Suppose p ∈ X , then the ideal I p ( X ) of real-analytic germs vanishing on X is generated by the components of a mapping f ( z, ¯ z ) . Let Σ p X be the germ at p of a complex-analytic subvariety given by the vanishing of z f ( z, ¯ p ) .Normally Σ p X is of the same complex codimension as is the real codimension of X . So if X is a real hypersurface, then Σ p X is usually a germ of a complex hypersurface. For ahypersurface we say X is Segre-degenerate at p if Σ p X is not a complex hypersurface, thatis, if Σ p X = ( C n , p ) .One of the main differences of real and complex varieties is that real varieties need not becoherent. A real-analytic subvariety is coherent if the sheaf of germs of real-analytic functionsvanishing on X is a coherent sheaf. Equivalently, X is coherent if it has a complexification,that is, a single variety that defines the complexification of all germs of X , or in yet otherwords, if for every p , representatives of the generators of I p ( X ) generate the ideals I q ( X ) forall nearby q . For the hypersurface case we prove the following. Theorem 1.1.
Let U ⊂ C n be open and X ⊂ U a real-analytic subvariety of codimension 1(a hypersurface). Let X [ n ] ⊂ X be the set of Segre-degenerate points. Then:(i) X [ n ] is a semianalytic set of dimension at most n − , which is locally contained in acomplex-analytic subvariety of (complex) dimension at most n − .(ii) If X is coherent, then X [ n ] is a complex-analytic subvariety of (complex) dimension atmost n − . The dimension of the complex subvariety may be smaller than n − . Example 6.1 givesa coherent hypersurface in C where X [ n ] is an isolated point. For noncoherent X , examplesexist for which X [ n ] is not a complex variety, or that are not even a real-analytic subvariety.In particular, the dimension of X [ n ] need not be even. Example 6.6 is a hypersurface in C such that (real) dimension of X [ n ] is 1. In Example 6.5, X [ n ] is only semianalytic and not areal-analytic subvariety.The Segre variety can be defined with respect to a specific defining function, or a neigh-borhood U of a point p . We call this variety Σ Uq X for q ∈ X . This Σ Up X can be takenas being defined by a representative of the generators of I p ( X ) for U small enough. For anoncoherent X it is possible that germ of Σ Uq X at q need not be the same as the germ Σ q X ,no matter how small U is and how close q is to p . It can even happen that there are regularpoints q arbitrarily close to p where Σ Uq X is singular (reducible) at q . See Example 6.4. If q is a regular point where X is generic (e.g. a hypersurface), the germ Σ q X is always regular.The point is that these germs Σ q X cannot be defined coherently by a single set of equationsfor a noncoherent subvariety.The results above are a special case of results for higher codimension. In general, the setof “Segre-degenerate points” would be points where the Segre variety is not of the expecteddimension. In some sense the main result of this paper is that for general X , we can stratify X into semianalytic sets by the dimension of the Segre variety. Theorem 1.2.
Let U ⊂ C n be open and X ⊂ U a real-analytic subvariety of dimension d < n (i.e. X = U ). Let X [ k ] ⊂ X be the set of points where the Segre variety is ofdimension k or higher. Then: EGRE-DEGENERATE POINTS FORM A SEMIANALYTIC SET 3 (i) For every k = 0 , , . . . , n , X [ k ] is a closed semianalytic subset of X , and X [ n ] is locallycontained in a complex-analytic subvariety of dimension at most n − d − .(ii) If X is coherent, then for every k = 0 , , . . . , n , X [ k ] is a closed real-analytic subvarietyof X , and X [ n ] is a complex-analytic subvariety of dimension at most n − d − . The sets X [ k ] are nested: X [ k +1] ⊂ X [ k ] . If X is of pure dimension d ≥ n , we find that X [ d − n ] = X . Then X [ n ] ⊂ · · · ⊂ X [ k ] ⊂ · · · ⊂ X [ d − n ] = X . If, furthermore, there exists aregular point of X where X is a generic submanifold, then X [ d − n +1] (the reasonable definitionof “Segre-degenerate points” in this case) is a semianalytic subset of X of dimension less than d , since where X is a generic submanifold the dimension of the Segre variety is necessarily d − n . We avoid defining the term Segre-degenerate for general X as the Segre varietiescan be degenerate in various ways; it is better to just talk about the sets X [ k ] or the sets X [ k ] \ X [ k +1] . In any case, since the sets X [ k ] are semianalytic, every reasonable definition of“ Segre-degenerate ” based on dimension leads to a semianalytic set.Remark that for k < n , the set X [ k ] is not necessarily complex even if it is a proper subsetof a coherent X , see Example 6.2.The structure of this paper is as follows. First we cover some preliminary results onsubvarieties and semianalytic sets in § 2. We introduce Segre varieties in the singular case in§ 3. In § 4, we prove the simpler results for the coherent case, and we cover the noncoherentcase in § 5. In § 6 we present some of the examples showing that the results are optimal andparticularly illustrating the degeneracy of the noncoherent case.2. Preliminaries
Definition 2.1.
Let U ⊂ R k (respectively U ⊂ C k ) be open. The set X ⊂ U is a real-analytic subvariety (resp. a complex-analytic subvariety ) of U if for each point p ∈ U , thereexists a neighborhood V ⊂ U of p and a set of real-analytic (resp. holomorphic) functions P ( V ) such that X ∩ V = { p ∈ V : f ( p ) = 0 for all f ∈ P ( V ) } . (1)Write X reg ⊂ X for the set of points which are regular , that is, X reg def = { p ∈ X : ∃ neighborhood V of p , such that V ∩ X is a real-analytic submanifold } . (2)The set of singular points is the complement: X sing def = X \ X reg . The dimension of X at p ∈ X reg , written as dim p X , is the real (resp. complex) dimension of the real-analytic (resp.complex) manifold at p . The dimension of X , written as dim X , is the maximum dimensionat any regular point. The dimension of X at p ∈ X sing is the minimum dimension of X ∩ V over all neighborhoods V of p . Define X ∗ def = { p ∈ X reg : dim p X = dim X } . (3)A variety or germ is irreducible if it cannot be written as a union of two proper subvarieties.Let I q ( X ) denote the ideal of germs ( f, q ) of functions that vanish on the germ ( X, q ) .Subvarieties are closed subsets of U . If a topology on X is required, we take the subspacetopology. Unlike in the complex case, a real-analytic subvariety can be a C k -manifold whilebeing singular as a subvariety. For example, x − y k +1 = 0 in R . Also, in the real case, theset of singular points need not be a subvariety and X ∗ need not equal to X reg . JIŘÍ LEBL
Definition 2.2.
For a set V (an open set in R n , or a subvariety), let S (cid:0) C ω ( V ) (cid:1) be thesmallest family of sets that is closed under finite unions, finite intersections, and complementsof sets of the form (cid:8) x ∈ V : f ( x ) ≥ (cid:9) (4)where f ∈ C ω ( V ) ( f real-analytic in V ).A set X ⊂ U is semianalytic (in U ) if for each p ∈ U , there is a neighborhood V of p suchthat X ∩ V ∈ S (cid:0) C ω ( V ) (cid:1) . Here U is an open set in R n , a subvariety, or an analytic space.Note that { x : f ( x ) ≤ } = { x : − f ( x ) ≥ } . Equality can be obtained by intersecting { x : f ( x ) ≥ } and { x : − f ( x ) ≥ } . Via complement we obtain sets of the form { x : f ( x ) > } and { x : f ( x ) = 0 } , thus we have all possible equalities and inequalities. Subvarieties aresemianalytic, but the family of semianalytic sets is much richer.If X is a complex-analytic subvariety, then X sing is a complex-analytic subvariety, while if X is only real-analytic, then X sing is only a semianalytic subset. Example 2.3.
The Whitney umbrella sx = y in R , using coordinates ( x, y, s ) (to avoidusing z as real), is a set where X sing is the set given by x = 0 , y = 0 , and s ≥ .It is a common misconception related to the subject of this paper to think that the setof singular points of a real subvariety X can be defined by the vanishing of the derivativesof functions that vanish on X . For a subvariety X defined near p , it is possible that dψ vanishes on some regular points of X arbitrarily near p for every function ψ defined near p such that ψ = 0 on X . Before proving this fact, let us prove a simple lemma. Lemma 2.4.
Suppose X ⊂ R k is a real cone, that is, if x ∈ X then λx ∈ X for all λ R .Suppose that X = { x : P ( x ) = 0 } for an irreducible homogeneous polynomial P (irreduciblein the ring of polynomials). Finally, suppose ( f, is a germ of a real-analytic function thatvanishes on X . Then ( f, is a multiple of the germ ( P, .In other words, I ( X ) is generated by the germ ( P, .Proof. The proof is standard: Write a representative f ( x ) = P ∞ k =0 f k ( x ) in terms of ho-mogeneous parts. Suppose x ∈ X , so f ( x ) = 0 . As λx ∈ X , then f ( λx ) ≡ . But then P ∞ k =0 f k ( λx ) = P ∞ k =0 λ k f k ( x ) is identically zero, meaning f k ( x ) = 0 for all k . The poly-nomial P generates the ideal of all polynomials vanishing on X and thus P divides all thepolynomials f k . Thence, the germ ( P, divides the germ ( f, . (cid:3) Example 2.5.
Let us give an example of a pure 2-dimensional real-analytic subvariety X ⊂ R with an isolated singularity at the origin, such that for any real-analytic definingfunction ψ of X near the origin, the set where both dψ and ψ vanish is a 1-dimensional subsetof X . Therefore the set where the derivative vanishes for the defining function is of largerdimension than the singular set, and dψ vanishes at some regular points. This subvarietywill be a useful example later (Example 6.4), and it is a useful example of a noncoherentsubvariety where coherence breaks not because of a smaller dimensional component.Let X be the subvariety of R in the coordinates ( x, y, s ) ∈ R : ( x + y ) − s x ( s − x ) = 0 . (5)We claim that X is as above. Despite the singularity being just the origin, for any real-analytic ψ defined near the origin that vanishes on X , we get dψ (0 , , s ) = 0 , so the derivativevanishes on (cid:8) ( x, y, s ) ∈ R : x = 0 , y = 0 (cid:9) = { } × { } × R ⊂ X. (6) EGRE-DEGENERATE POINTS FORM A SEMIANALYTIC SET 5
As this example will be useful for Segre varieties, we prove the claim in detail. Thesubvariety in R defined by ( x + y ) − x (1 − x ) = 0 is irreducible algebraically (thepolynomial is irreducible). In fact, it is a compact submanifold of R . It is nonsingular near0 because if we write ( x + y ) = x (1 − x ) and take the third root we get ( x + y ) = x √ − x. (7)Near the origin we can solve for x using the implicit function theorem.Homogenize ( x + y ) − x (1 − x ) with s to get the set X in R given by (5). The set X is a cone with an isolated singularity; it is a cone over a manifold. By Lemma 2.4, if ψ vanishes on X , then ψ = (cid:0) ( x + y ) − s x ( s − x ) (cid:1) ϕ. (8)In other words, on X , dψ must vanish where the derivative of ( x + y ) − s x ( s − x ) vanishes.3. Segre varieties
Consider subvarieties of C n ∼ = R n . Let U ⊂ C n be open and X ⊂ U a real-analyticsubvariety. Write U c = { z : ¯ z ∈ U } for the complex conjugate. Let ι ( z ) = ( z, ¯ z ) be theembedding of C n into the “diagonal” in C n × C n . Denote by X U the smallest complex-analyticsubvariety of U × U c such that ι ( X ) ⊂ X U . It is standard that there exists a small enough U (see below) such that X U ∩ ι ( C n ) = ι ( X ) . Let σ : C n × C n → C n × C n denote the involution σ ( z, w ) = ( ¯ w, ¯ z ) . Note that the “diagonal” ι ( C n ) is the fixed set of σ . Proposition 3.1.
Let U ⊂ C n be open and X ⊂ U be a real-analytic subvariety. Then σ ( X U ) = X U .Proof. σ ( X U ) is an defined by vanishing of anti-holomorphic functions, and therefore byholomorphic functions, and so it is a complex-analytic subvariety. As X is fixed by σ wehave X ⊂ σ ( X U ) ∩ X U , and the result follows as X U is the smallest subvariety. (cid:3) The ideal I p ( X ) can be generated by the real and imaginary parts of the generators of theideal of germs of holomorphic functions defined at ( p, ¯ p ) in the complexification that vanishon the germ of ι ( X ) at ( p, ¯ p ) . Call this ideal I p ( X ) .Given a germ of a real-analytic subvariety ( X, p ) , denote by X p the smallest germ of acomplex-analytic subvariety of (cid:0) C n × C n , ( p, ¯ p ) (cid:1) that contains the image of ( X, p ) by ι . Thegerm X p is called the complexification of ( X, p ) . It is not hard to see that the irreduciblecomponents of ( X, p ) correspond to the irreducible components of X p ; if ( X, p ) is irreducible,so is X p . In the theory of real-analytic subvarieties, X U would not be called a complexification of X unless (cid:0) X U , ( p, ¯ p ) (cid:1) = X p for all p ∈ X , and that cannot always be achieved.As we will need a specific neighborhood of a point often we make the following definition. Definition 3.2.
Let X ⊂ U be a real-analytic subvariety of dimension d of an open set U ⊂ C n . We say U is good for X at p ∈ X if the following conditions are satisfied:(i) U is connected.(ii) The real dimension of ( X, p ) is d and the complex dimension of X p and X U is also d .(iii) There exists a real-analytic function ψ : U → R k whose complexification converges in U × U c , whose zero set is X , and the germ ( ψ, p ) generates I p ( X ) .(iv) X U ∩ ι ( C n ) = ι ( X ) .(v) (cid:0) X U , ( p, ¯ p ) (cid:1) = X p . JIŘÍ LEBL (vi) The irreducible components of X U correspond in a one-to-one fashion to the irreduciblecomponents of the germ X p .If U ′ ⊂ U is good for X ∩ U ′ at p we say simply that U ′ is good for X at p . Proposition 3.3.
Suppose U ⊂ C n is open, X ⊂ U is a real-analytic subvariety, and p ∈ X .Then there exists a neighborhood U ′ ⊂ U of p such that U ′ is good for X at p .Furthermore, for any neighborhood W of p , there exists a neighborhood W ′ ⊂ W of p , thatis good for X at p .Proof. The idea is standard (see e.g. [8]), but let us sketch a proof. The main difficultyis mostly notational. Take the germ I p ( X ) of complexified functions that vanish on vanishon the germ of ι ( X ) at ( p, ¯ p ) . Note that I p ( X ) is closed under the conjugation taking ψ to ψ ◦ σ , that is, ψ ( z, ζ ) to ¯ ψ ( ζ , z ) . It is generated by a finite set of functions f , . . . , f k ,which are all defined in some polydisc ∆ × ∆ c centered at p . The real and imaginary partsof these functions also generate an ideal, and this ideal must be equal to I p ( X ) . We can alsoassume that ∆ is small enough so that all the components of the subvariety V defined by f , . . . , f k go through ( p, ¯ p ) (in other words V is the smallest subvariety of ∆ containing thegerm of ι ( X ) at ( p, ¯ p ) ). Similarly, make ∆ small enough so that the real and imaginary partsof f , . . . , f k restricted to the diagonal, give the subvariety X ∩ ∆ all of whose componentsgo through p . We can take ∆ to also be small enough so that all components of X p havedistinct representatives in ∆ . The set ∆ must be our U ′ . (cid:3) Definition 3.4.
Suppose U ⊂ C n is open and X ⊂ U a real-analytic subvariety. The Segrevariety of X at p ∈ U relative to U is the set Σ Up X def = (cid:8) z ∈ U : ( z, ¯ p ) ∈ X U (cid:9) . (9)If U ′ ⊂ U , we write Σ U ′ p X for Σ U ′ p ( X ∩ U ′ ) .When U ′ is good for X at p ∈ X , define the germ Σ p X def = (cid:0) Σ U ′ p X, p (cid:1) . (10)Define X [ k ] def = (cid:8) z ∈ U : dim Σ z X ≥ k (cid:9) , (11) X U [ k ] def = (cid:8) z ∈ U : dim z Σ Uz X ≥ k (cid:9) . (12)The germ Σ p X is well-defined by the proposition. First, there exists a good neighbor-hood of p , and any smaller good neighborhood of p would give us the same germ of thecomplexification at p .If X is an irreducible hypersurface, X is Segre degenerate at p ∈ X if Σ p X = ( C n , p ) ,that is, if p ∈ X [ n ] . A point p is Segre degenerate relative to U if dim p Σ Up X = n , that is, if p ∈ X U [ n ] . A key point of this paper that these two notions can be different. We will seethat X U [ n ] is always a complex subvariety and contains X [ n ] , and the two are not necessarilyequal even for a small enough U . They may not be even of the same dimension.For general dimension d set, we will simply talk about the sets X [ k ] and we will not makea judgement on what is the best definition for the word “Segre-degenerate.” EGRE-DEGENERATE POINTS FORM A SEMIANALYTIC SET 7
A submanifold is generic (see [2]) at a point if given the defining functions ρ , . . . , ρ k , theholomorphic parts of the exterior derivative, ∂ρ , . . . , ∂ρ k , are linearly independent. Equiv-alently, it is generic if in some local coordinates vanishing at p it is Im w = r ( z, ¯ z, Re w ) , . . . , Im w k = r k ( z, ¯ z, Re w ) , (13)with r j and its derivative vanishing at . A nonsingular hypersurface is always generic. Proposition 3.5.
Suppose X is a real-analytic submanifold of C n of dimension d (so codi-mension n − d ) and p ∈ X .(i) dim Σ p X ≤ d . In particular, if d < n , then X [ n ] = ∅ .(ii) If X is generic at p , then Σ p X is a germ of a complex submanifold and dim Σ p X = d − n .Proof. We start with the generic case. Using the defining functions above, k = 2 n − d , wenote that if we plug in ¯ w = 0 and ¯ z = 0 , we get k linearly independent defining equationsfor a complex submanifold.If X is not generic, then we can write down similar equations but solving for real andimaginary parts. Since these could conceivably be the real and imaginary parts of the samevariable, we may only get k = n − d independent equations, so the dimension of Σ p X couldbe as high as n − n − d = d . (cid:3) If X is regular at p but not generic, the germ Σ p X could possibly be singular and thedimension may vary as p moves on the submanifold. See Example 6.3.Let us collect some basic properties of Segre varieties in the singular case. Proposition 3.6.
Let X ⊂ U ⊂ C n be a real-analytic subvariety of dimension d and p ∈ X .Then(i) Σ p X ⊂ (cid:0) Σ Up X, p (cid:1) .(ii) dim Σ p X ≥ d − n .(iii) X [ k ] ⊂ X U [ k ] for all k .(iv) If U is good for X at p and d ≤ n , then dim Σ Up X < n .(v) If d ≤ n , then X [ n ] = ∅ .(vi) q ∈ Σ q X if and only if q ∈ X , and so X [ k ] ⊂ X for all k = 0 , , , . . . , n .(vii) If U is good for X at p , q ∈ Σ Uq X if and only if q ∈ X , and so X U [ k ] ⊂ X for all k = 0 , , , . . . , n .Proof. If U ′ ⊂ U then Σ U ′ p ⊂ Σ Up , as any analytic function defined on U that vanishes on X is an analytic function on U ′ that vanishes on X ∩ U ′ . Parts (i) and (iii) follow.For part (ii), the complexification X p has dimension d . Let U ′ be good for X at p . Thegerm Σ p X is the germ at p of the intersection of X U ′ and C n × { ¯ p } . The codimension of X U ′ at ( p, ¯ p ) in C n × C n is n − d , and the codimension of C n × { ¯ p } is n . Hence their intersectionis of codimension at most n − d or dimension n − (3 n − d ) = d − n .To see (iv) first note that if d < n , then it is impossible for Σ Uq X to be of dimension n as itis a subvariety of X U , which is of dimension d < n . If d = n , then without loss of generalitysuppose that ( X, p ) is irreducible. As U is good for X at p , then X U is also irreducible. Bydimension, as X U is of dimension n and dim Σ p X = n , then U × { ¯ p } would be an irreduciblecomponent of X U . By symmetry (applying σ ), { p } × U c is also an irreducible component of X U . This is a contradiction as X U is irreducible.Then (v) follows from (iv) by considering a small enough good neighborhood of every q ∈ X . JIŘÍ LEBL
For (vi), if q ∈ X , then q ∈ Σ q X , since ψ ( q, ¯ q ) = 0 for any germ of a function at q thatvanishes on X . If q / ∈ X , then clearly Σ q X = ∅ . So X [ k ] ⊂ X .For (vii), again if q ∈ X , then it must be that q ∈ Σ Uq X . Similarly, for a good U , we have X U = ι ( X ) , and so q ∈ Σ Uq X means that q ∈ X . (cid:3) The point of this paper is that even for arbitrarily small neighborhoods U of p (even goodfor X at p ) and a q ∈ U that is arbitrarily close to p , it is possible that (Σ Up X, q ) = Σ q X. (14)That is, unless X is coherent. Let us focus on X [ n ] for a moment. It is possible that for allneighborhoods U of a point p ∈ X , X [ n ] = X U [ n ] . (15)The set X U [ n ] is rather well-behaved. Proposition 3.7.
Let X ⊂ U ⊂ C n be a real-analytic subvariety of dimension d < n , and U is good for X at some p ∈ X . Then X U [ n ] is a complex-analytic subvariety of dimension atmost n − d − . In particular, X [ n ] is contained in a complex-analytic subvariety of dimensionat most n − d − .Proof. Without loss of generality, suppose that ( X, p ) is irreducible. The variety X U is fixedby the involution σ . In other words, ( z, ¯ w ) ∈ X U if and only if ( w, ¯ z ) ∈ X U . So if q ∈ X U [ n ] ,then ( z, ¯ q ) ∈ X U for all z ∈ U , and therefore ( q, ¯ z ) ∈ X U for all z ∈ U . In particular, q ∈ Σ Uz X for all z ∈ U . As Σ Uz X is a complex-analytic subvariety, generically of dimension d − n , then X U [ n ] is a complex-analytic subvariety of dimension at most d − n .The only way that X U [ n ] could be of dimension d − n is if all the varieties Σ Uz X containeda fixed complex-analytic subvariety V of dimension d − n . This means that V × C n ⊂ X U and C n × V c ⊂ X U (by applying σ ). By dimension, these are components of X U . Since weassumed that ( X, p ) is irreducible, so is X p and so is X U if U is good for X at p , and weobtain a contradiction. Hence, X U [ n ] must be of dimension at most d − n − . (cid:3) Coherent varieties
A real-analytic subvariety is coherent if the sheaf of germs of real-analytic functions van-ishing on X is a coherent sheaf. The fundamental fact about coherent subvarieties is thatthey possess a global complexification. That is, if X is coherent, then there exists a complex-analytic subvariety X of some neighborhood of X in C n × C n such that X ∩ ι ( C n ) = X and (cid:0) X , ( p, ¯ p ) (cid:1) is equal to X p , the complexification of the germ ( X, p ) at every p ∈ X . See [8]. Lemma 4.1.
Let X ⊂ U ⊂ C n be a real-analytic subvariety. If X ⊂ U is a coherent and U is good for X at p ∈ X , then Σ q X = (Σ Uq X, q ) for all q ∈ X . In particular, X U [ n ] = X [ n ] .Proof. Since X is coherent, we have a global complexification X and hence X U = X ∩ U .In particular this is true for any good neighborhood U ′ ⊂ U of any point q ∈ X , so Σ q X =(Σ U ′ q X, q ) = (Σ Uq X, q ) .As X U [ n ] ⊂ X and it is the set where (Σ Uq X, q ) are of dimension n , we find that it is equalto the set where Σ q X is of dimension n . In other words, X U [ n ] = X [ n ] . (cid:3) We can now prove the theorem for coherent subvarieties. The following theorem impliesthe coherent part of Theorem 1.2 for k = n , and hence the coherent part of Theorem 1.1. EGRE-DEGENERATE POINTS FORM A SEMIANALYTIC SET 9
Theorem 4.2.
Let U ⊂ C n be open and X ⊂ U be a coherent real-analytic subvariety ofdimension d < n . Then X [ n ] is a complex-analytic subvariety of dimension at most d − n − .Proof. It is sufficient to work in a good neighborhood of some point, without loss of generalityassume that U is good for some p ∈ X . Apply the lemma and Proposition 3.7. (cid:3) For general k , we have the following theorem, which finishes the coherent case of Theo-rem 1.2 for k < n . That is, for every k , the X [ k ] sets are subvarieties of X for coherent X .These subvarieties no longer need to be complex-analytic. Theorem 4.3.
Let U ⊂ C n be open and X ⊂ U be a coherent real-analytic subvariety. Thenfor every k = 0 , , . . . , n , X [ k ] is a real-analytic subvariety of X . A generic submanifold has the Segre variety of the least possible dimension. Let X be anirreducible coherent subvariety of dimension d . If X reg is generic at some point, then Σ q X isof (the minimum possible) dimension d − n somewhere, and hence the Segre degenerate set,the set where the dimension of Σ q X is higher, that is, X [ d − n +1] , is according to this theorema real-analytic subvariety of X . Proof.
It is a local result and so without loss of generality assume that U is good for X atsome p ∈ X . Let ( z, ξ ) be the complexified variables of C n × C n . Consider the projection π ( z, ξ ) = ξ defined on X U . The Segre variety Σ Uz X is (identified with) the fiber π − (cid:0) π ( z, ¯ z ) (cid:1) .The dimension of the germ Σ z X is the dimension at ( z, ¯ z ) of π − (cid:0) π ( z, ¯ z ) (cid:1) as X is coherent.For any integer k , the set V k = n ( z, ξ ) ∈ X U : dim ( z,ξ ) π − (cid:0) π ( z, ξ ) (cid:1) ≥ k o (16)is a complex-analytic subvariety of X U (see e.g. Theorem 9F in chapter 7 of Whitney [13]).Then V k ∩ { ξ = ¯ z } = ι ( X [ k ] ) is a real-analytic subvariety of ι ( X ) . (cid:3) The set of Segre degenerate points is semianalytic
It is rather simple to prove that X [ k ] is always closed (in classical, not Zariski, topology). Proposition 5.1.
Let U ⊂ C n be open and X ⊂ U be a real-analytic subvariety. Then q dim Σ q X is an upper semi-continuous function on X . In particular, for every k , X [ k ] is closed.Proof. Let p ∈ X be some point and let U be good for X at p and follow the construction inthe proof of Theorem 4.3, that is let X U and π be as before. The Segre variety Σ Uz X is thefiber π − (cid:0) π ( z, ¯ z ) (cid:1) . For all z ∈ X , Σ z X is a subset (possibly proper as X is not coherent) ofthe germ (Σ Uz X, z ) , and so dim z Σ Uz X ≥ dim Σ z X . As U is good for X at p , (Σ Up X, p ) = Σ p X .As the sets V k ∩ ι ( X ) are closed, dim Σ p X = (Σ Up X, p ) is bounded below by dimensions of (Σ Uq X, q ) for all sufficiently nearby q , and these are in turn bounded below by dim Σ q X . (cid:3) We need some results about semianalytic subsets. We are going to use normalization on X U and so we need to prove that semianalytic sets are preserved under finite holomorphicmappings. The key point in that proof is the following theorem on projection of semialgebraicsets extended to handle certain semianalytic sets. Theorem 5.2 (Łojasiewicz–Tarski–Seidenberg (see [3, 10])) . Let A be a ring of real-valuedfunctions on an set U , and let π : U × R m → U be the projection.If X ∈ S (cid:0) A [ t , . . . , t m ] (cid:1) , then π ( X ) ∈ S ( A ) . Complex-analytic subvarieties are preserved under finite (or just proper) holomorphicmaps. Real semialgebraic sets are preserved under all real polynomial maps. On the otherhand real-analytic subvarieties or semianalytic sets are not preserved by finite or properreal-analytic maps. But, as long as the map is holomorphic and finite, semianalytic setsare preserved. Here is an intuitive useful argument of why this is expected: Map for-ward the complexification of a real-analytic subvariety by the complexification of the map ( z, ¯ z ) (cid:0) f ( z ) , ¯ f (¯ z ) (cid:1) , which is still finite, so it maps the complexification to a complex-analytic subvariety. So the image of a real-analytic subvariety of dimension d via a finiteholomorphic map is contained in a real-analytic subvariety of dimension d . To get equalitywe need to go to semianalytic sets: Think of z z as the map and the real line as thereal-analytic subvariety. The holomorphicity is required as the complexification of a finitereal-analytic map need not be finite (simple example: z z ¯ z + i ( z + ¯ z ) ). Lemma 5.3.
Let
V, W be complex analytic spaces, let S ⊂ V be a semianalytic set, and f : V → W be a finite holomorphic map. Then f ( S ) is semianalytic of the same dimensionas S .Proof. Without loss of generality, assume that f ( V ) = W . Furthermore, since the map isfinite, and finite unions of semianalytic sets are semianalytic, assume that V, W are actualcomplex-analytic subvarieties by working locally in some chart, and in general we can justassume we are working in an arbitrarily small neighborhood of the origin ∈ V , and that f (0) = 0 . Suppose V is a subvariety of some neighborhood U ⊂ C n , and W is a subvarietyof some open set U ′ ⊂ C m . By adding components to f equal to the defining functions of V (and thus possibly increasing m ) we can assume without loss of generality that f : U → C m is a finite map on U and not just V .Consider the graph Γ f of f in U × C m . Because f is finite, the projection of Γ f to C m isfinite. In particular, the variety Γ f can be defined by functions that are polynomials in thefirst n variables (in fact polynomials in the first n variables and a few of the last m variablesdepending on the codimension of W = f ( V ) in C m ). Let z = x + iy denote the first n variables, and ξ denote the last m variables. The variety Γ f as a real subvariety is definedby functions that are polynomials in x and y .Also assume that U is small enough so that S is defined by real-analytic functions in U , that is, S ∈ S (cid:0) C ω ( U ) (cid:1) . The set S corresponds to a semianalytic set e S ⊂ Γ f . Theset e S is defined by functions defined in some U × U ′ , suppose ϕ is one of these functions.The subvariety Γ f is defined by polynomials in x and y , so we find Weierstrass polynomialsin every one of x and y with coefficients real-analytic functions in ξ that are in the real-analytic ideal for Γ f at (0 , . Since adding anything in the ideal does not change ϕ whereit matters (on Γ f ), we can divide by these polynomials and find a remainder ψ , which is apolynomial in x and y such that ψ = ϕ on Γ f . In other words, e S ∈ S (cid:0) C ω ( U ′ )[ x, y ] (cid:1) . By theŁojasiewicz–Tarski–Seidenberg theorem, the projection of e S to U ′ is semianalytic.The fact that the dimension is preserved follows from f being finite. (cid:3) Remark . The lemma is not true if f is not holomorphic and finite. If f is proper but notholomorphic, the best we can conclude is that f ( S ) is subanalytic as long as we also assumethat S is precompact. Our task would be easier if we only desired to prove that X [ k ] aresubanalytic. EGRE-DEGENERATE POINTS FORM A SEMIANALYTIC SET 11
The proof that X [ k ] is semianalytic for non-coherent subvarieties is similar to Theorem 4.3,but we work on the normalization of the complex variety X U . Theorem 5.5.
Let U ⊂ C n be open and X ⊂ U be a real-analytic subvariety. Then forevery k = 0 , , . . . , n , X [ k ] is a closed semianalytic subset of X .Proof. Again, it is a local result so without loss of generality assume that U is good for X at some p ∈ X and suppose that X is irreducible at p and that X is of dimension d .Consider h : Y → X U the normalization of X U . There are two reasons why X U is not thecomplexification at some point q . For points z arbitrarily near q , either the set X is of lowerdimension at z , or there are multiple irreducible components of the germ (cid:0) X U , ( z, ¯ z ) (cid:1) .Let X ∗ denote the relative closure in U of the set of points of dimension d . The set X \ X ∗ is semianalytic, and so locally near any q ∈ X it is possible to write X = X ∗ ∪ X ′ for X ′ a real-analytic subvariety of lower dimension (possibly empty) defined in a neighborhoodof q . Suppose for induction that X ′ [ k ] is semianalytic. Then X ′ [ k ] \ X ∗ = X [ k ] \ X ∗ is alsosemianalytic (in a neighborhood of q ). In other words, it remains to prove that X [ k ] ∩ X ∗ issemianalytic.Let X = h − (cid:0) ι ( X ∗ ) (cid:1) , and note that this is a closed semianalytic subset of Y of dimension d , although it can have points of various dimensions. Therefore, take X = X ∗ to be theclosure (in Y ) of the nonsingular points of X of dimension d . It is clear that h ( X ) = ι ( X ∗ ) .Let ( z, ξ ) be the complexified variables of C n × C n , where X U lives. Consider the projection π ( z, ξ ) = ξ defined on X U . The Segre variety Σ Uz X is the fiber π − (cid:0) π ( z, ¯ z ) (cid:1) , but the germat ( z, ¯ z ) may contain other components, so we pull back to Y .Let η be the variable on Y and we pull back via h as ( π ◦ h ) − (cid:0) π ◦ h ( η ) (cid:1) . The space Y is normal and so the germ ( Y , η ) is irreducible for all η . Near some η ∈ X , the set X is a totally-real subset of Y of dimension d , and hence ( Y , η ) , which is irreducible and ofdimension d , contains ( X , η ) is then the smallest complex subvariety containing ( X , η ) .The germ of the complexification of X at h ( η ) has as its components the images of ( Y , η ′ ) via h for all η ′ ∈ h − ( h ( η )) ∩ X . These images must be contained in the complexificationand as h ( X ) = ι ( X ∗ ) , their union is the entire complexification of X at h ( η ) . We thus needto consider the sets W k = n η ∈ Y : dim η ( π ◦ h ) − (cid:0) π ◦ h ( η ) (cid:1) ≥ k o , (17)which are again complex analytic. We are interested in the sets X ∩ W k , which are semian-alytic, and we have just proved above that h ( X ∩ W k ) = ι ( X [ k ] ) . As h is finite and X ∩ W k is semianalytic, we are finished. (cid:3) Examples of Segre variety degeneracies
Example 6.1.
The set of Segre-degenerate points of a coherent hypersurface in C n can bea complex subvariety of dimension strictly less than n − . Let X ⊂ C in coordinates ( z, w, ξ ) ∈ C be given by z ¯ z + w ¯ w − ξ ¯ ξ = 0 . (18)The set of regular points is everything except the origin, so only the origin can be Segre-degenerate, and for this subvariety, it is, as the above equation generates the ideal I ( X ) byLemma 2.4. So X [3] = { } , which is of dimension n − . Example 6.2.
For a higher codimensional subvariety, the set X [ k ] for k < n is generally notcomplex. Clearly if k ≤ d − n , then X [ k ] = X and X is not necessarily complex. But even forhigher k less than n , the set need not be complex. Let X ⊂ C in coordinates ( z, w, ξ ) ∈ C be given by z ¯ z − w ¯ w = 0 , Im ξ = 0 . (19)The subvariety X is -dimensional and coherent. It is easy to see that X [1] = X , X [2] = { z =0 , w = 0 , Im ξ = 0 } , and X [3] = ∅ . The set X [2] is not complex. Example 6.3.
A submanifold may be Segre-degenerate, if it is CR singular. Let ( z, w ) bethe coordinates in C and consider the manifold X given by w = z ¯ z. (20)As this is a complex equation, to find the generators of the ideal we must take the real andimaginary parts, or equivalently, also consider the conjugate of the equation, ¯ w = z ¯ z . Forpoints where z = 0 , the Segre variety is just the trivial point, so zero dimensional. But at thepoint (0 , the Segre variety is the complex line { w = 0 } . In other words, X [0] = X \{ (0 , } , X [1] = { (0 , } , and X [2] = ∅ .Similarly, the Segre variety of a submanifold can be singular if the manifold is CR singular.Let ( z, w, ξ ) be coordinates in C and consider X given by w = z + ¯ z + ξ + ¯ ξ . (21)The Segre subvariety at the origin Σ X is the pair of complex lines given by { w = 0 , ( z + iξ )( z − iξ ) = 0 } . Example 6.4.
Consider Example 2.5, that is ( x + y ) − s x ( s − x ) = 0 and extend it to C using z = x + iy and w = s + it . In other words, we use X × R if X is the variety of theprevious example. That is, let X in ( z, w ) ∈ C be given by f ( z, w, ¯ z, ¯ w ) = ( z ¯ z ) − (Re w ) (Re z ) (Re w − Re z ) = 0 . (22)Similarly as in Example 2.5, this f generates the ideal at I ( X ) , its derivatives vanish when z = 0 , but X is regular outside of { z = 0 , Re w = 0 } . So there are regular (hypersurface,thus generic) points of X where the complexified f defines a singular subvariety. That is,regular points of X where the corresponding X U is singular for any neighborhood U of .For these points q , for any U , Σ q X is regular, but Σ Uq X is singular at q . In particular, Σ q X ( (Σ Uq , q ) . (23)So Σ q X is just one component of the germ (Σ Uq , q ) . Example 6.5.
The set of Segre-degenerate points of a hypersurface need not be a subvarietyfor noncoherent X . Let X ⊂ C in coordinates ( z, w, ξ ) ∈ C be given by z ¯ z − ( ξ + ¯ ξ ) w ¯ w = 0 . (24)The set is reminiscent of the Whitney umbrella. The set U = C is a good neighborhood for X at . The set of Segre-degenerate points with respect to U (actually any neighborhood U of the origin), is X U [3] = { w = z = 0 } , that is, a one dimensional complex line. However,where Re ξ < , then the variety X is locally just the line { w = z = 0 } . Therefore, thevariety is a real manifold of dimension 2 (complex manifold of dimension 1). At such points EGRE-DEGENERATE POINTS FORM A SEMIANALYTIC SET 13 Σ p X is just one dimensional and hence such points are not in X [3] (not Segre-degenerate).Hence, X [3] = { ( z, w, ξ ) ∈ X : w = z = 0 , Re ξ ≥ } (25)and this is not a subvariety, it is a semianalytic set. Example 6.6.
Let us construct the promised noncoherent hypersurface in C where the set X [ n ] of Segre-degenerate points is not complex, in fact, it is a real line.Let X ⊂ C in coordinates ( z, w, ξ ) ∈ C be given by ψ = w ¯ w (Re ξ ) + 4(Re z )(Re ξ ) w ¯ w + 4(Re z ) z ¯ z = 0 . (26)The function is irreducible as a polynomial and homogeneous and thus ( X, is irreducibleas a germ of a real-analytic subvariety.The set where dψ = 0 is given by Re z = 0 , w = 0 , and this set lies in X . Therefore, { dψ = 0 } ⊂ X is 3-real dimensional. However the singular set X sing is 2-dimensional givenby Re z = 0 , w = 0 , and Re ξ = 0 . Let us prove this fact. For simplicity let z = x + iy and ξ = s + it and assume s = 0 . Solve for w ¯ w as w ¯ w = x (cid:18) − s ± s p s − sx ( x + y ) (cid:19) . (27)When the ± = − and s = 0 , we can solve for x by implicit function theorem and thesubvariety has a regular point there. If ± = + and s = 0 , then we claim that there is nosolution except x = 0 , s = 0 , w = 0 . We must check a few possibilities. If x > and s > ,then s p s − sx ( x + y ) < s , and as w ¯ w must be positive there are no such real solutions.Similarly for every other sign combination. That means that the only solution when s = 0 is when the ± = + . So X is regular when Re ξ = s = 0 . Similarly, it is not difficult toshow that X is singular at points where Re z = 0 , w = 0 , Re ξ = 0 : For example, at suchpoints, were they regular, the Re z = 0 hyperplane and the Re z = − Re ξ hyperplane wouldboth have to be tangent as their intersections with X are singular (both reducible). That isimpossible for a regular point.Since ψ generates the ideal at the origin, it is easy to see that X U [ n ] = { z = 0 , w = 0 } nearthe origin for any good neighborhood U of the origin. As X [ n ] ⊂ X U [ n ] and X [ n ] ⊂ X sing , wecan see that X [ n ] ⊂ { z = 0 , w = 0 , Re ξ = 0 } . Since the defining function does not dependon Im ξ , all the points of the set { z = 0 , w = 0 , Re ξ = 0 } are in X [ n ] or none of them are.The origin is definitely Segre-degenerate as ψ is the generator of the ideal there, and thus X [ n ] = { z = 0 , w = 0 , Re ξ = 0 } . So the set X [ n ] where X is Segre-degenerate is of realdimension 1.In other words:(i) dim X sing = 2 .(ii) { df = 0 } ∩ X is 3 real-dimensional for every real-analytic germ f vanishing on X .(iii) The set of Segre-degenerate points X [ n ] is a real one-dimensional line.(iv) The set of Segre-degenerate points relative to U , X U [ n ] , is a complex one-dimensionalline at the origin for every good neighborhood U of the origin, and X U [ n ] ∩ X reg = ∅ . References [1] Janusz Adamus, Serge Randriambololona, and Rasul Shafikov,
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